BW Transform of Thue-Morse Words Chapter 94 in 125 problems book. Speaker: @kgoto Book Reading Seminar #18, March 7, 2022 1 1

We call BW-string of BW Transform The Burrows-Wheeler transform of length- string is the word composed of the last letters of the sorted conjugates of . BW(x) n x x BW(x) x 2 2

Thue-Morse Words Thue-Morse morphism from -th Thue-Morse words for μ {a, b}∗ μ(a) = ab μ(b) = ba n τ ​ n τ ​ = 0 a τ ​ = 1 ab τ ​ = 2 abba τ ​ = 3 abbabaab τ ​ = n μ(τ ​ ) n−1 n > 0 Today's topic The recursive structure of BW-string of Thue-Morse words , where , BW (τ ​ ) = n+1 b ​ a n−1 BW (τ ​ ) n n−1 = a b = b a 3 3

Observation : matrix whose rows are sorted rotations of is split to three parts : top rows : middle rows, which corresponds to : bottom rows S ​ n+1 2 × n+1 2n+1 τ ​ n+1 S ​ n+1 T ​ n+1 2n−1 M ​ n+1 2n μ(S ​ ) n B ​ n+1 2n−1 4 4

Observation and occur in , respectively. in produces in , then such occurs in is composed of the rotation by splitting the middle of such If , then If , then Since thre is no occurrence of , then does not contain , and the bottom of starts with Top of starts with a b 2n−1 τ ​ n b τ ​ n ba τ ​ n+1 ba 2n−1 τ ​ n+1 T ​ n+1 ba s < 1 s ​ 2 μ(s ​ ) < 1 μ(s ​ ) 2 s = bt aμ(t)b < μ(t)ba < baμ(t) bbb T ​ n+1 ababa T ​ n+1 abaa M ​ n+1 abab 5 5

Observation Symmetically, is composed of the rotation by splitting the middle of produced by in starts with and ends with starts with and ends with The last characters of are flipped characters of ones of B ​ n+1 ab a τ ​ n T ​ n+1 a b B ​ n+1 b a M ​ n+1 S ​ n+1 Conclusion , where , BW (τ ​ ) = n+1 b ​ a n−1 BW (τ ​ ) n n−1 = a b = b a 6 6

Reference Brlek, S., Frosini, A., Mancini, I., Pergola, E. and Rinaldi, S., 2019, July. Burrows-Wheeler transform of words defined by morphisms. In International Workshop on Combinatorial Algorithms (pp. 393-404). Springer, Cham. https://link.springer.com/chapter/10.1007/978-3-030-25005-8_32 7 7